Is null vector always linearly dependent?

If $\{\vec v_1,\vec v_2, \cdots, \vec v_n\}$ are linearly independent $c_1 \vec v_1+c_2 \vec v_2+\cdots+c_n \vec v_n= \vec 0$ iff $c_1=c_2=\cdots=c_n=0$. Considering $\vec v_n=\vec 0$, we can get $c_1 \vec v_1+c_2 \vec v_2+\cdots+c_n \vec v_n= \vec 0$ by setting $c_1=c_2=\cdots=c_{n-1}=0$ and taking any $c_n \neq 0$. So by definition, any set of vectors that contain the zero vector is linearly dependent.

It is exactly as you say: in any vector space, the null vector belongs to the span of any vector.